Finite Group Databasebeta

← #129695

Subgroups — #129695

Order 512.

552 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 116 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2986C211yesmaximal, characteristic
2#11642C212yesmaximal, complemented
3#11639C212yesmaximal, complemented
4#11460C211yesmaximal, characteristic
5#11459C211yesmaximal, characteristic
6C2×#533C411yescharacteristic
7#2986—24no—
8#2986—24no—
9C2×#446—21no—
10C2×#446C411yescharacteristic
11C2×#446C2211yescharacteristic
12C2×C8×Q8—21no—
13C2×C8×Q8C411yescharacteristic
14C2×C8×Q8C2211yescharacteristic
15C2×#433—21no—
16C2×#433—21no—
17C2×#431C411yescharacteristic
18C2×#431C2211yescharacteristic
19#2163C2212yes—
20#2162C2212yes—
21#1338—24no—
22Q8⋊C16—24no—
23#1322—21no—
24#1322—21no—
25#1314—21no—
26#1314—21no—
27#533—22no—
28#533—22no—
29#533—22no—
30#533—22no—
31C2×C4⋊Q8C811yescharacteristic
32C2×C4⋊Q8C811yescharacteristic
33C2×#125—21no—
34C2×C22⋊Q8—21no—
35C2×C22⋊Q8C811yescharacteristic
36C2×C22⋊Q8C811yescharacteristic
37C2×C4×Q8—21no—
38C2×C4×Q8C2×C411yescharacteristic
39C2×C4×Q8C2×C411yescharacteristic
40C2×#117—21no—
41C22×C42C2×C411yescharacteristic
42#446—28no—
43#446—22no—
44#446—22no—
45#446—22no—
46#446D414yes—
47C8×Q8—28no—
48C8×Q8—22no—
49C8×Q8—22no—
50C8×Q8—22no—
51C8×Q8D414yes—
52#433—28no—
53#433—28no—
54#431—22no—
55#431—22no—
56#431—22no—
57#431—22no—
58C2×C4⋊C8—21no—
59C2×C4⋊C8—21no—
60C2×C4⋊C8—21no—
61C2×C4⋊C8—21no—
62C2×C4⋊C8C2×C411yescharacteristic
63C2×C4⋊C8C2×C411yescharacteristic
64C2×C4⋊C8D411yescharacteristic
65C2×C4⋊C8D411yescharacteristic
66C2×C22⋊C8—21no—
67C2×C22⋊C8D411yescharacteristic
68C2×C22⋊C8D411yescharacteristic
69C2×#97—21no—
70C2×#97—21no—
71C2×C4×C8—21no—
72C2×C4×C8—21no—
73C2×C4×C8—21no—
74C2×C4×C8C2×C411yescharacteristic
75C2×C4×C8C2311yescharacteristic
76C4⋊C16—22no—
77C4⋊C16—22no—
78C22⋊C16—22no—
79C22⋊C16—22no—
80#346—22no—
81C4×C16—22no—
82C22×Q8—21no—
83C22×Q8C2×C811yescharacteristic
84C22×Q8C2×C811yescharacteristic
85C23×C4C22⋊C411yescharacteristic
86C23×C4C2×C811yescharacteristic
87C23×C4M1611yescharacteristic
88C22×C8—21no—
89C22×C8—21no—
90C22×C8C22⋊C411yescharacteristic
91C22×C8C22⋊C411yescharacteristic
92C22×C8C2×D411yescharacteristic
93C22×C8C2×D411yescharacteristic
94C4⋊Q8—22no—
95C4⋊Q8—22no—
96C4⋊Q8—22no—
97C4⋊Q8—22no—
98#125—24no—
99#125—24no—
100C22⋊Q8—24no—
101C22⋊Q8—24no—
102C22⋊Q8—22no—
103C22⋊Q8—22no—
104C22⋊Q8—22no—
105C22⋊Q8—22no—
106C4×Q8—24no—
107C4×Q8—24no—
108C4×Q8—21no—
109C4×Q8—21no—
110C4×Q8—21no—
111C4×Q8—21no—
112C4×Q8C22⋊C412yes—
113C4×Q8C22⋊C412yes—
114C4×Q8M1612yes—
115C4×Q8M1612yes—
116#117—24no—
117#117—24no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4C2×C811yescharacteristic
126C2×C4⋊C4C2×C811yescharacteristic
127C2×C22⋊C4—21no—
128C2×C22⋊C4—21no—
129C2×C42—21no—
130C2×C42—21no—
131C2×C42—21no—
132C2×C42—21no—
133C2×C42—21no—
134C2×C42—21no—
135C2×C42C22⋊C411yescharacteristic
136C2×C42C2×C811yescharacteristic
137C2×C42M1611yescharacteristic
138C2×C42C22×C411yescharacteristic
139C2×C16—42no—
140C2×C16—42no—
141C4⋊C8—24no—
142C4⋊C8—24no—
143C4⋊C8—24no—
144C4⋊C8—24no—
145C4⋊C8—22no—
146C4⋊C8—22no—
147C4⋊C8—21no—
148C4⋊C8—21no—
149C4⋊C8C22⋊C412yes—
150C4⋊C8C22⋊C412yes—
151C22⋊C8—24no—
152C22⋊C8—22no—
153C22⋊C8—22no—
154#97—24no—
155#97—24no—
156C4×C8—24no—
157C4×C8—24no—
158C4×C8—24no—
159C4×C8—21no—
160C4×C8—21no—
161C4×C8C2×D412yes—
162C4×C8C2×D412yes—
163C24C22⋊C811yescharacteristic
164C2×Q8—24no—
165C2×Q8—21no—
166C2×Q8—21no—
167C2×Q8—42no—
168C2×Q8—42no—
169C2×Q8—42no—
170C2×Q8—42no—
171C2×Q8C22⋊C812yes—
172C2×Q8C22⋊C812yes—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—22no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—42no—
188C22×C4C22⋊C811yescharacteristic
189C22×C4C22⋊C811yescharacteristic
190C22×C4C22⋊C811yescharacteristic
191C22×C4C2×C22⋊C411yescharacteristic
192C22×C4C22×C811yescharacteristic
193C22×C4C2×M1611yescharacteristic
194C2×C8—21no—
195C2×C8—22no—
196C2×C8—21no—
197C2×C8—22no—
198C2×C8—41no—
199C2×C8—41no—
200C2×C8—42no—
201C2×C8—42no—
202C2×C8—41no—
203C2×C8—41no—
204C2×C8#12012yes—
205C2×C8#12012yes—
206C4⋊C4—24no—
207C4⋊C4—24no—
208C4⋊C4—24no—
209C4⋊C4—24no—
210C4⋊C4—24no—
211C4⋊C4—21no—
212C4⋊C4—21no—
213C4⋊C4—22no—
214C4⋊C4—22no—
215C4⋊C4—22no—
216C4⋊C4—22no—
217C4⋊C4C22⋊C812yes—
218C4⋊C4C22⋊C812yes—
219C22⋊C4—24no—
220C22⋊C4—24no—
221C42—21no—
222C42—22no—
223C42—22no—
224C42—21no—
225C42—21no—
226C42—21no—
227C42—21no—
228C42—21no—
229C42—22no—
230C42—22no—
231C42C2×C22⋊C411yescharacteristic
232C42C2×C22⋊C411yescharacteristic
233C42C2×M1611yescharacteristic
234C42C2×M1611yescharacteristic
235C16—82no—
236C16—82no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—41no—
241C23—41no—
242C23C2×C22⋊C811yescharacteristic
243Q8—24no—
244Q8—42no—
245Q8—42no—
246Q8—48no—
247Q8—42no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—42no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4#35012yes—
277C2×C4#37911yescharacteristic
278C2×C4C2×C22⋊C811yescharacteristic
279C2×C4C2×C22⋊C811yescharacteristic
280C2×C4#40711yescharacteristic
281C2×C4#40711yescharacteristic
282C2×C4#40711yescharacteristic
283C8—22no—
284C8—22no—
285C8—41no—
286C8—41no—
287C8—42no—
288C8—42no—
289C22—21no—
290C22—21no—
291C22—41no—
292C22—41no—
293C22—41no—
294C22—81no—
295C22—81no—
296C22#177911yescharacteristic
297C22C2×#35011yescharacteristic
298C22#211711yescharacteristic
299C4—21no—
300C4—21no—
301C4—22no—
302C4—21no—
303C4—21no—
304C4—41no—
305C4—41no—
306C4—42no—
307C4—42no—
308C4—42no—
309C4—42no—
310C4#179611yescharacteristic
311C4#179611yescharacteristic
312C4#211512yes—
313C2—21no—
314C2—21no—
315C2—81no—
316C2#832311yescharacteristic
317C2#1268211yescharacteristic
318C2#1268311yescharacteristic
subgroupG/H
centerC2×C4#379
commutatorC22×C4C22×C8
Frattini, cosocleC2×C4×C8C23
socleC22#1779
prime radicalC24C22⋊C8
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #11642C2semidirect
3. #11639C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#29862
#116422
#116392
#114602
#114592